Present Value of Ordinary Annuity

by Obaidullah Jan, ACA, CFA

An annuity is a finite stream of equal cash flows that occur after equal interval. There are two types of annuities: ordinary annuity and annuity due. The (ordinary) annuity (which is also referred to as just annuity) is an annuity in which each periodic cash flow occurs at the end of each period.

Many financial products are in fact annuities, for example bonds. Most bonds pay fixed coupon payments after equal interval from their issue date to their maturity date. Bonds are priced by discounting those coupon payments and the final terminal redemption value to time 0 based on the market interest rates.

Formula

One way to find the present value of an annuity is to manually discount each cash flow in the stream using the following equation and then summing all the component present values to find the present value of the annuity.

$$ PV=\frac{PMT}{\left(1+\frac{r}{m}\right)^{n\times m}} $$

Where PMT is the periodic payment in annuity, r is the annual interest rate, n is the number of years between time 0 and the relevant payment date and m is the number of annuity payments per year.

Alternatively, we can calculate the present value of the ordinary annuity directly using the following formula:

$$ PV=PMT\times\frac{1\ -\ \left(1+\frac{r}{m}\right)^{-n\times m}}{\frac{r}{m}} $$

The same calculation can be carried out using Excel PV function. PV function syntax is PV(rate, nper, pmt, [fv], [type]). Where rate is the periodic interest rate (i.e. annual percentage rate divided by number of payments per year), nper is the total number of payments, pmt is the amount of payment, [fv] is an optional argument allowing us to specify if there is any final balloon payment. [type] is an optional argument that specifies whether the annuity is an ordinary annuity or an annuity due. By default, Excel assumes the annuity to be an ordinary annuity.

Example

Let’s use the present value of an annuity formulas to find price of treasury bond that has 2 years till maturity. The bond has a par value of $100 and coupon rate of 3% thereby paying $1.5 coupon after each six-month period. The market yield on the bond is 2.9%. Calculate the price of the bond.

The price of the bond equals the present value all bond cash flows, both coupon payment and the final redemption value. The coupon payments form an ordinary annuity because they are equal and occur after equal interval (i.e. 6 months) while the final redemption value i.e. $100 paid back at the bond maturity date is a single sum.

We can manually calculate the bond price by individually discounting each coupon payment and the redemption value as follows:

$$ PV\\=\frac{$1.5}{\left(1+\frac{2.9\%}{2}\right)^1}+\frac{$1.5}{\left(1+\frac{2.9\%}{2}\right)^2}+\frac{$1.5}{\left(1+\frac{2.9\%}{2}\right)^3}\\+\frac{$1.5}{\left(1+\frac{2.9\%}{2}\right)^4}+\frac{$100}{\left(1+\frac{2.9\%}{2}\right)^4}\\=$5.79+$94.40\\=$100.19 $$

The first four terms on the right-hand side forms an annuity and the final term is the present value of a single sum.

The present value of the coupon payments can be calculated as follows:

$$ PV\\=$1.5\times\frac{1\ -\ \left(1+\frac{2.9%}{2}\right)^{-2\times2}}{\frac{2.90%}{2}}+\frac{$100}{\left(1+\frac{2.9%}{2}\right)^4}\\=$5.79+$94.40\\=$100.19 $$

If you want to calculate present value using Excel in the above situation, you need to enter the following function: PV(2.9%/2,4,-1.5,-100,0).